relation to Fuchsian equation

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21: 7.11 Relations to Other Functions

§7.11 Relations to Other Functions

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Incomplete Gamma Functions and Generalized Exponential Integral

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7.11.1 $$\mathrm{erf}z=\frac{1}{\sqrt{\pi }}\gamma (\frac{1}{2},z^2),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions

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22: 15.17 Mathematical Applications

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§15.17(i) Differential Equations

►This topic is treated in §§15.10 and 15.11. … ► … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …

23: 13.6 Relations to Other Functions

§13.6 Relations to Other Functions

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§13.6(iv) Parabolic Cylinder Functions

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§13.6(v) Orthogonal Polynomials

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Laguerre Polynomials

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§13.6(vi) Generalized Hypergeometric Functions

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24: 16.24 Physical Applications

… ►For an extension to two-loop integrals see Moch et al. (2002). ►

§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

… ►The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner $\mathit{6}j$ symbols. …

25: Simon Ruijsenaars

… ►His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. … ►

28 Mathieu Functions and Hill’s Equation

26: 14.31 Other Applications

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§14.31(i) Toroidal Functions

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§14.31(ii) Conical Functions

… ►The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. ►

§14.31(iii) Miscellaneous

►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

27: 13.27 Mathematical Applications

… ►Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives. … ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).

28: 10.39 Relations to Other Functions

§10.39 Relations to Other Functions

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Elementary Functions

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Parabolic Cylinder Functions

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions and Hypergeometric Function

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29: 31.12 Confluent Forms of Heun’s Equation

… ►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i). … ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. … ►

Biconfluent Heun Equation

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Triconfluent Heun Equation

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30: 9.16 Physical Applications

… ►The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). … ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler–Tricomi equation (Landau and Lifshitz (1987)). An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). … ►Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics. …